[X] Tyrion's heir is his own to choose or make, but a betrothal between your Houses could strengthen your reign. Perhaps Ser Kevan's daughter Janei to your brother Bran?
[X] Send North 10,000 men of the Westerlands under a lord recommended by Tyrion Lannister for his loyalty and military acumen (Crakehall? Daven Lannister?). Send North 10,000 men of the Reach under a lord recommended by Randyll Tarly for his loyalty and military acumen. Send North 10,000 men of the Stormlands under a lord recommended by Davos Seaworth for his loyalty and military acumen. Send 250 men with Greatjon Umber who is to command the entire force. Have their lords order them to follow the Greatjon's commands until they reach North, and upon joining with your father, follow Lord Stark's commands. There are reports of a huge wildling army coming to cross the Wall; their mission is to stop them. All the while, advise the Greatjon to be wary of the Golden Company landing on the east coast and the Ironborn attacking the west coast.
[X] Make for Storm's End. You cannot lose the Baratheons or their claim.
-[x] Send some aid to the North to help them hold until you can get up there yourself
[X] Tyrion's likely successor is his cousin Martyn, a boy close to Sansa's age. Betrothing him to Arya would bind the West more strongly to your House.
[X] As King you have the power to legitimize whoever you please. Tommen is a sweet boy whatever his parentage and a betrothal to Arya would provide you with a malleable and easily led heir of the West.